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Theorem isnumbasgrplem2 38463
Description: If the (to be thought of as disjoint, although the proof does not require this) union of a set and its Hartogs number supports a group structure (more generally, a cancellative magma), then the set must be numerable. (Contributed by Stefan O'Rear, 9-Jul-2015.)
Assertion
Ref Expression
isnumbasgrplem2 ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) → 𝑆 ∈ dom card)

Proof of Theorem isnumbasgrplem2
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 basfn 16208 . . 3 Base Fn V
2 ssv 3825 . . 3 Grp ⊆ V
3 fvelimab 6482 . . 3 ((Base Fn V ∧ Grp ⊆ V) → ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) ↔ ∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))))
41, 2, 3mp2an 684 . 2 ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) ↔ ∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
5 harcl 8712 . . . . . 6 (har‘𝑆) ∈ On
6 onenon 9065 . . . . . 6 ((har‘𝑆) ∈ On → (har‘𝑆) ∈ dom card)
75, 6ax-mp 5 . . . . 5 (har‘𝑆) ∈ dom card
8 xpnum 9067 . . . . 5 (((har‘𝑆) ∈ dom card ∧ (har‘𝑆) ∈ dom card) → ((har‘𝑆) × (har‘𝑆)) ∈ dom card)
97, 7, 8mp2an 684 . . . 4 ((har‘𝑆) × (har‘𝑆)) ∈ dom card
10 ssun1 3978 . . . . . . . 8 𝑆 ⊆ (𝑆 ∪ (har‘𝑆))
11 simpr 478 . . . . . . . 8 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
1210, 11syl5sseqr 3854 . . . . . . 7 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ⊆ (Base‘𝑥))
13 fvex 6428 . . . . . . . 8 (Base‘𝑥) ∈ V
1413ssex 5001 . . . . . . 7 (𝑆 ⊆ (Base‘𝑥) → 𝑆 ∈ V)
1512, 14syl 17 . . . . . 6 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ∈ V)
167a1i 11 . . . . . 6 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (har‘𝑆) ∈ dom card)
17 simp1l 1255 . . . . . . . 8 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑥 ∈ Grp)
18123ad2ant1 1164 . . . . . . . . 9 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑆 ⊆ (Base‘𝑥))
19 simp2 1168 . . . . . . . . 9 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑎𝑆)
2018, 19sseldd 3803 . . . . . . . 8 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑎 ∈ (Base‘𝑥))
21 ssun2 3979 . . . . . . . . . . 11 (har‘𝑆) ⊆ (𝑆 ∪ (har‘𝑆))
2221, 11syl5sseqr 3854 . . . . . . . . . 10 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (har‘𝑆) ⊆ (Base‘𝑥))
23223ad2ant1 1164 . . . . . . . . 9 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → (har‘𝑆) ⊆ (Base‘𝑥))
24 simp3 1169 . . . . . . . . 9 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑐 ∈ (har‘𝑆))
2523, 24sseldd 3803 . . . . . . . 8 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑐 ∈ (Base‘𝑥))
26 eqid 2803 . . . . . . . . 9 (Base‘𝑥) = (Base‘𝑥)
27 eqid 2803 . . . . . . . . 9 (+g𝑥) = (+g𝑥)
2826, 27grpcl 17750 . . . . . . . 8 ((𝑥 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑥) ∧ 𝑐 ∈ (Base‘𝑥)) → (𝑎(+g𝑥)𝑐) ∈ (Base‘𝑥))
2917, 20, 25, 28syl3anc 1491 . . . . . . 7 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → (𝑎(+g𝑥)𝑐) ∈ (Base‘𝑥))
30 simp1r 1256 . . . . . . 7 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
3129, 30eleqtrd 2884 . . . . . 6 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → (𝑎(+g𝑥)𝑐) ∈ (𝑆 ∪ (har‘𝑆)))
32 simplll 792 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑥 ∈ Grp)
3322ad2antrr 718 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → (har‘𝑆) ⊆ (Base‘𝑥))
34 simprl 788 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑐 ∈ (har‘𝑆))
3533, 34sseldd 3803 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑐 ∈ (Base‘𝑥))
36 simprr 790 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑑 ∈ (har‘𝑆))
3733, 36sseldd 3803 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑑 ∈ (Base‘𝑥))
3812ad2antrr 718 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑆 ⊆ (Base‘𝑥))
39 simplr 786 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑎𝑆)
4038, 39sseldd 3803 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑎 ∈ (Base‘𝑥))
4126, 27grplcan 17797 . . . . . . 7 ((𝑥 ∈ Grp ∧ (𝑐 ∈ (Base‘𝑥) ∧ 𝑑 ∈ (Base‘𝑥) ∧ 𝑎 ∈ (Base‘𝑥))) → ((𝑎(+g𝑥)𝑐) = (𝑎(+g𝑥)𝑑) ↔ 𝑐 = 𝑑))
4232, 35, 37, 40, 41syl13anc 1492 . . . . . 6 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → ((𝑎(+g𝑥)𝑐) = (𝑎(+g𝑥)𝑑) ↔ 𝑐 = 𝑑))
43 simplll 792 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑥 ∈ Grp)
4412ad2antrr 718 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑆 ⊆ (Base‘𝑥))
45 simprr 790 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑑𝑆)
4644, 45sseldd 3803 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑑 ∈ (Base‘𝑥))
47 simprl 788 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑎𝑆)
4844, 47sseldd 3803 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑎 ∈ (Base‘𝑥))
4922ad2antrr 718 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → (har‘𝑆) ⊆ (Base‘𝑥))
50 simplr 786 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑏 ∈ (har‘𝑆))
5149, 50sseldd 3803 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑏 ∈ (Base‘𝑥))
5226, 27grprcan 17775 . . . . . . 7 ((𝑥 ∈ Grp ∧ (𝑑 ∈ (Base‘𝑥) ∧ 𝑎 ∈ (Base‘𝑥) ∧ 𝑏 ∈ (Base‘𝑥))) → ((𝑑(+g𝑥)𝑏) = (𝑎(+g𝑥)𝑏) ↔ 𝑑 = 𝑎))
5343, 46, 48, 51, 52syl13anc 1492 . . . . . 6 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → ((𝑑(+g𝑥)𝑏) = (𝑎(+g𝑥)𝑏) ↔ 𝑑 = 𝑎))
54 harndom 8715 . . . . . . 7 ¬ (har‘𝑆) ≼ 𝑆
5554a1i 11 . . . . . 6 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → ¬ (har‘𝑆) ≼ 𝑆)
5615, 16, 16, 31, 42, 53, 55unxpwdom3 38454 . . . . 5 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆* ((har‘𝑆) × (har‘𝑆)))
57 wdomnumr 9177 . . . . . 6 (((har‘𝑆) × (har‘𝑆)) ∈ dom card → (𝑆* ((har‘𝑆) × (har‘𝑆)) ↔ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆))))
589, 57ax-mp 5 . . . . 5 (𝑆* ((har‘𝑆) × (har‘𝑆)) ↔ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆)))
5956, 58sylib 210 . . . 4 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ≼ ((har‘𝑆) × (har‘𝑆)))
60 numdom 9151 . . . 4 ((((har‘𝑆) × (har‘𝑆)) ∈ dom card ∧ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆))) → 𝑆 ∈ dom card)
619, 59, 60sylancr 582 . . 3 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ∈ dom card)
6261rexlimiva 3213 . 2 (∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)) → 𝑆 ∈ dom card)
634, 62sylbi 209 1 ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) → 𝑆 ∈ dom card)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wrex 3094  Vcvv 3389  cun 3771  wss 3773   class class class wbr 4847   × cxp 5314  dom cdm 5316  cima 5319  Oncon0 5945   Fn wfn 6100  cfv 6105  (class class class)co 6882  cdom 8197  harchar 8707  * cwdom 8708  cardccrd 9051  Basecbs 16188  +gcplusg 16271  Grpcgrp 17742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2379  ax-ext 2781  ax-rep 4968  ax-sep 4979  ax-nul 4987  ax-pow 5039  ax-pr 5101  ax-un 7187
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2593  df-eu 2611  df-clab 2790  df-cleq 2796  df-clel 2799  df-nfc 2934  df-ne 2976  df-ral 3098  df-rex 3099  df-reu 3100  df-rmo 3101  df-rab 3102  df-v 3391  df-sbc 3638  df-csb 3733  df-dif 3776  df-un 3778  df-in 3780  df-ss 3787  df-pss 3789  df-nul 4120  df-if 4282  df-pw 4355  df-sn 4373  df-pr 4375  df-tp 4377  df-op 4379  df-uni 4633  df-int 4672  df-iun 4716  df-br 4848  df-opab 4910  df-mpt 4927  df-tr 4950  df-id 5224  df-eprel 5229  df-po 5237  df-so 5238  df-fr 5275  df-se 5276  df-we 5277  df-xp 5322  df-rel 5323  df-cnv 5324  df-co 5325  df-dm 5326  df-rn 5327  df-res 5328  df-ima 5329  df-pred 5902  df-ord 5948  df-on 5949  df-lim 5950  df-suc 5951  df-iota 6068  df-fun 6107  df-fn 6108  df-f 6109  df-f1 6110  df-fo 6111  df-f1o 6112  df-fv 6113  df-isom 6114  df-riota 6843  df-ov 6885  df-oprab 6886  df-mpt2 6887  df-om 7304  df-1st 7405  df-2nd 7406  df-wrecs 7649  df-recs 7711  df-rdg 7749  df-1o 7803  df-oadd 7807  df-omul 7808  df-er 7986  df-map 8101  df-en 8200  df-dom 8201  df-sdom 8202  df-oi 8661  df-har 8709  df-wdom 8710  df-card 9055  df-acn 9058  df-slot 16192  df-base 16194  df-0g 16421  df-mgm 17561  df-sgrp 17603  df-mnd 17614  df-grp 17745  df-minusg 17746
This theorem is referenced by:  isnumbasabl  38465  isnumbasgrp  38466
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